Imaging a Black Hole using only Closure Quantities

In this tutorial, we will create a preliminary reconstruction of the 2017 M87 data on April 6 using closure-only imaging. This tutorial is a general introduction to closure-only imaging in Comrade. For an introduction to simultaneous image and instrument modeling, see Stokes I Simultaneous Image and Instrument Modeling

Introduction to Closure Imaging

The EHT is the highest-resolution telescope ever created. Its resolution is equivalent to roughly tracking a hockey puck on the moon when viewing it from the earth. However, the EHT is also a unique interferometer. For one, the data it produces is incredibly sparse. The array is formed from only eight geographic locations around the planet, each with its unique telescope. Additionally, the EHT observes at a much higher frequency than typical interferometers. As a result, it is often difficult to directly provide calibrated data since the source model can be complicated. This implies there can be large instrumental effects often called gains that can corrupt our signal. One way to deal with this is to fit quantities that are independent of gains. These are often called closure quantities. The types of closure quantities are briefly described in Introduction to the VLBI Imaging Problem.

In this tutorial, we will do closure-only modeling of M87 to produce preliminary images of M87.

To get started, we will load Comrade

using Comrade




using Pyehtim

  Activating project at `~/work/Comrade.jl/Comrade.jl/examples`

For reproducibility we use a stable random number genreator

using StableRNGs
rng = StableRNG(123)

StableRNGs.LehmerRNG(state=0x000000000000000000000000000000f7)

Load the Data

To download the data visit https://doi.org/10.25739/g85n-f134 To load the eht-imaging obsdata object we do:

obs = ehtim.obsdata.load_uvfits(joinpath(dirname(pathof(Comrade)), "..", "examples", "SR1_M87_2017_096_lo_hops_netcal_StokesI.uvfits"))

Python: <ehtim.obsdata.Obsdata object at 0x7fdfb236b790>

Now we do some minor preprocessing:

• Scan average the data since the data have been preprocessed so that the gain phases are coherent.
• Add 1% systematic noise to deal with calibration issues that cause 1% non-closing errors.

obs = scan_average(obs).add_fractional_noise(0.015)

Python: <ehtim.obsdata.Obsdata object at 0x7fdfb236b760>

Now, we extract our closure quantities from the EHT data set.

dlcamp, dcphase  = extract_table(obs, LogClosureAmplitudes(;snrcut=3), ClosurePhases(;snrcut=3))

(EHTObservation{Float64,Comrade.EHTLogClosureAmplitudeDatum{Float64}, ...}
  source: M87
  mjd: 57849
  frequency: 2.27070703125e11
  bandwidth: 1.856e9
  stations: [:AA, :AP, :AZ, :JC, :LM, :PV, :SM]
  nsamples: 128
, EHTObservation{Float64,Comrade.EHTClosurePhaseDatum{Float64}, ...}
  source: M87
  mjd: 57849
  frequency: 2.27070703125e11
  bandwidth: 1.856e9
  stations: [:AA, :AP, :AZ, :JC, :LM, :PV, :SM]
  nsamples: 152
)

Build the Model/Posterior

For our model, we will be using an image model that consists of a raster of point sources, convolved with some pulse or kernel to make a ContinuousImage object with it Comrade's. generic image model.

function sky(θ, metadata)
    (;fg, c, σimg) = θ
    (;K, meanpr, grid, cache) = metadata
    # Construct the image model we fix the flux to 0.6 Jy in this case
    cp = meanpr .+ σimg.*c.params
    rast = ((1-fg))*K(to_simplex(CenteredLR(), cp))
    img = IntensityMap(rast, grid)
    m = ContinuousImage(img, cache)
    # Add a large-scale gaussian to deal with the over-resolved mas flux
    g = modify(Gaussian(), Stretch(μas2rad(250.0), μas2rad(250.0)), Renormalize(fg))
    return m + g
end

sky (generic function with 1 method)

Now, let's set up our image model. The EHT's nominal resolution is 20-25 μas. Additionally, the EHT is not very sensitive to a larger field of views; typically, 60-80 μas is enough to describe the compact flux of M87. Given this, we only need to use a small number of pixels to describe our image.

npix = 32
fovxy = μas2rad(150.0)

7.27220521664304e-10

Now, we can feed in the array information to form the cache

grid = imagepixels(fovxy, fovxy, npix, npix)
buffer = IntensityMap(zeros(npix,npix), grid)
cache = create_cache(NFFTAlg(dlcamp), buffer, BSplinePulse{3}())

VLBISkyModels.NUFTCache{VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}, NFFT.NFFTPlan{Float64, 2, 1}, Vector{ComplexF64}, BSplinePulse{3}, KeyedArray{Float64, 2, NamedDimsArray{(:X, :Y), Float64, 2, Matrix{Float64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}}}(VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}(NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}(1, 4, 2.0, :kaiser_bessel, AbstractNFFTs.POLYNOMIAL, true, false, true, 0x00000000), [-4.405690154666661e9 787577.6145833326 … -5.999801315555549e9 -15551.297851562484; -4.523017159111106e9 -1.6838098888888871e6 … 3.059254300444441e9 118294.64453124987]), NFFTPlan with 274 sampling points for an input array of size(32, 32) and an output array of size(274,) with dims 1:2, ComplexF64[0.7014670924816703 - 0.519511453845776im, 0.9999999862106403 - 6.398624051637031e-5im, 0.6947985314160843 - 0.5233374146500747im, 0.9328845732113544 - 0.16336912604935844im, 0.9329421134238738 + 0.16331744240563614im, 0.6948519110447875 + 0.5233083088067162im, 0.7925669922789302 + 0.40478845348863063im, 0.9999999864389528 - 6.519719206334107e-5im, 0.9340555032742773 - 0.1520910238460368im, 0.6896791530194316 - 0.5261728587493705im  …  0.8376666586657548 + 0.1784956664688869im, 0.9196690775195503 + 0.09173305645579018im, 0.9135584686897926 + 0.29199716871629755im, 0.9135610039654034 + 0.2919905620278203im, 0.8899638273254515 + 0.28719490128071723im, 0.9568100109264257 + 0.09814170760184636im, 0.8899622135777404 + 0.287201492222944im, 0.9925668796456906 + 0.0027765725149323534im, 0.8376650833595861 - 0.17848899851834446im, 0.9999999999247222 + 7.335331210317892e-6im], BSplinePulse{3}(), [0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0])

Now we need to specify our image prior. For this work we will use a Gaussian Markov Random field prior

using VLBIImagePriors, Distributions, DistributionsAD

Since we are using a Gaussian Markov random field prior we need to first specify our mean image. For this work we will use a symmetric Gaussian with a FWHM of 50 μas

fwhmfac = 2*sqrt(2*log(2))
mpr = modify(Gaussian(), Stretch(μas2rad(50.0)./fwhmfac))
imgpr = intensitymap(mpr, grid)

2-dimensional KeyedArray(NamedDimsArray(...)) with keys:
↓   X ∈ 32-element LinRange{Float64,...}
→   Y ∈ 32-element LinRange{Float64,...}
And data, 32×32 NamedDimsArray(::Matrix{Float64}, (:X, :Y)):
                 (-3.52247e-10)  …  (3.29522e-10)  (3.52247e-10)
 (-3.52247e-10)     6.37537e-8        1.32434e-7     6.37537e-8
 (-3.29522e-10)     1.32434e-7        2.751e-7       1.32434e-7
 (-3.06796e-10)     2.62014e-7        5.44273e-7     2.62014e-7
 (-2.84071e-10)     4.93724e-7        1.0256e-6      4.93724e-7
 (-2.61345e-10)     8.86092e-7   …    1.84065e-6     8.86092e-7
 (-2.38619e-10)     1.51463e-6        3.14629e-6     1.51463e-6
    ⋮                            ⋱    ⋮            
  (2.15894e-10)     2.46586e-6        5.12225e-6     2.46586e-6
  (2.38619e-10)     1.51463e-6        3.14629e-6     1.51463e-6
  (2.61345e-10)     8.86092e-7   …    1.84065e-6     8.86092e-7
  (2.84071e-10)     4.93724e-7        1.0256e-6      4.93724e-7
  (3.06796e-10)     2.62014e-7        5.44273e-7     2.62014e-7
  (3.29522e-10)     1.32434e-7        2.751e-7       1.32434e-7
  (3.52247e-10)     6.37537e-8        1.32434e-7     6.37537e-8

Now since we are actually modeling our image on the simplex we need to ensure that our mean image has unit flux

imgpr ./= flux(imgpr)

meanpr = to_real(CenteredLR(), Comrade.baseimage(imgpr))
metadata = (;meanpr,K=CenterImage(imgpr), grid, cache)

(meanpr = [-7.55422122563378 -6.823167558636962 … -6.823167558636962 -7.55422122563378; -6.823167558636962 -6.092113891640146 … -6.092113891640146 -6.823167558636962; … ; -6.823167558636962 -6.092113891640146 … -6.092113891640146 -6.823167558636962; -7.55422122563378 -6.823167558636962 … -6.823167558636962 -7.55422122563378], K = VLBIImagePriors.CenterImage{Matrix{Float64}, Tuple{Int64, Int64}}([0.9944957386363662 -0.005326704545454548 … 0.005326704545454572 0.005504261363636381; -0.005326704545454548 0.9948393969941349 … 0.005160603005865089 0.005326704545454565; … ; 0.005326704545454572 0.005160603005865089 … 0.9948393969941348 -0.005326704545454546; 0.005504261363636381 0.005326704545454565 … -0.005326704545454546 0.9944957386363638], (32, 32)), grid = GriddedKeys{(:X, :Y)}
	X: LinRange{Float64}(-3.5224744018114725e-10, 3.5224744018114725e-10, 32)
	Y: LinRange{Float64}(-3.5224744018114725e-10, 3.5224744018114725e-10, 32)
, cache = VLBISkyModels.NUFTCache{VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}, NFFT.NFFTPlan{Float64, 2, 1}, Vector{ComplexF64}, BSplinePulse{3}, KeyedArray{Float64, 2, NamedDimsArray{(:X, :Y), Float64, 2, Matrix{Float64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}}}(VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}(NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}(1, 4, 2.0, :kaiser_bessel, AbstractNFFTs.POLYNOMIAL, true, false, true, 0x00000000), [-4.405690154666661e9 787577.6145833326 … -5.999801315555549e9 -15551.297851562484; -4.523017159111106e9 -1.6838098888888871e6 … 3.059254300444441e9 118294.64453124987]), NFFTPlan with 274 sampling points for an input array of size(32, 32) and an output array of size(274,) with dims 1:2, ComplexF64[0.7014670924816703 - 0.519511453845776im, 0.9999999862106403 - 6.398624051637031e-5im, 0.6947985314160843 - 0.5233374146500747im, 0.9328845732113544 - 0.16336912604935844im, 0.9329421134238738 + 0.16331744240563614im, 0.6948519110447875 + 0.5233083088067162im, 0.7925669922789302 + 0.40478845348863063im, 0.9999999864389528 - 6.519719206334107e-5im, 0.9340555032742773 - 0.1520910238460368im, 0.6896791530194316 - 0.5261728587493705im  …  0.8376666586657548 + 0.1784956664688869im, 0.9196690775195503 + 0.09173305645579018im, 0.9135584686897926 + 0.29199716871629755im, 0.9135610039654034 + 0.2919905620278203im, 0.8899638273254515 + 0.28719490128071723im, 0.9568100109264257 + 0.09814170760184636im, 0.8899622135777404 + 0.287201492222944im, 0.9925668796456906 + 0.0027765725149323534im, 0.8376650833595861 - 0.17848899851834446im, 0.9999999999247222 + 7.335331210317892e-6im], BSplinePulse{3}(), [0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]))

In addition we want a reasonable guess for what the resolution of our image should be. For radio astronomy this is given by roughly the longest baseline in the image. To put this into pixel space we then divide by the pixel size.

beam = beamsize(dlcamp)
rat = (beam/(step(grid.X)))

5.326336637737519

To make the Gaussian Markov random field efficient we first precompute a bunch of quantities that allow us to scale things linearly with the number of image pixels. This drastically improves the usual N^3 scaling you get from usual Gaussian Processes.

crcache = MarkovRandomFieldCache(meanpr)

VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}(sparse([1, 2, 32, 33, 993, 1, 2, 3, 34, 994  …  31, 991, 1022, 1023, 1024, 32, 992, 993, 1023, 1024], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 29, 29, 29, 29, 29, 30, 30, 30, 30, 30, 31, 31, 31, 31, 31, 32, 32, 32, 32, 32, 33, 33, 33, 33, 33, 34, 34, 34, 34, 34, 35, 35, 35, 35, 35, 36, 36, 36, 36, 36, 37, 37, 37, 37, 37, 38, 38, 38, 38, 38, 39, 39, 39, 39, 39, 40, 40, 40, 40, 40, 41, 41, 41, 41, 41, 42, 42, 42, 42, 42, 43, 43, 43, 43, 43, 44, 44, 44, 44, 44, 45, 45, 45, 45, 45, 46, 46, 46, 46, 46, 47, 47, 47, 47, 47, 48, 48, 48, 48, 48, 49, 49, 49, 49, 49, 50, 50, 50, 50, 50, 51, 51, 51, 51, 51, 52, 52, 52, 52, 52, 53, 53, 53, 53, 53, 54, 54, 54, 54, 54, 55, 55, 55, 55, 55, 56, 56, 56, 56, 56, 57, 57, 57, 57, 57, 58, 58, 58, 58, 58, 59, 59, 59, 59, 59, 60, 60, 60, 60, 60, 61, 61, 61, 61, 61, 62, 62, 62, 62, 62, 63, 63, 63, 63, 63, 64, 64, 64, 64, 64, 65, 65, 65, 65, 65, 66, 66, 66, 66, 66, 67, 67, 67, 67, 67, 68, 68, 68, 68, 68, 69, 69, 69, 69, 69, 70, 70, 70, 70, 70, 71, 71, 71, 71, 71, 72, 72, 72, 72, 72, 73, 73, 73, 73, 73, 74, 74, 74, 74, 74, 75, 75, 75, 75, 75, 76, 76, 76, 76, 76, 77, 77, 77, 77, 77, 78, 78, 78, 78, 78, 79, 79, 79, 79, 79, 80, 80, 80, 80, 80, 81, 81, 81, 81, 81, 82, 82, 82, 82, 82, 83, 83, 83, 83, 83, 84, 84, 84, 84, 84, 85, 85, 85, 85, 85, 86, 86, 86, 86, 86, 87, 87, 87, 87, 87, 88, 88, 88, 88, 88, 89, 89, 89, 89, 89, 90, 90, 90, 90, 90, 91, 91, 91, 91, 91, 92, 92, 92, 92, 92, 93, 93, 93, 93, 93, 94, 94, 94, 94, 94, 95, 95, 95, 95, 95, 96, 96, 96, 96, 96, 97, 97, 97, 97, 97, 98, 98, 98, 98, 98, 99, 99, 99, 99, 99, 100, 100, 100, 100, 100, 101, 101, 101, 101, 101, 102, 102, 102, 102, 102, 103, 103, 103, 103, 103, 104, 104, 104, 104, 104, 105, 105, 105, 105, 105, 106, 106, 106, 106, 106, 107, 107, 107, 107, 107, 108, 108, 108, 108, 108, 109, 109, 109, 109, 109, 110, 110, 110, 110, 110, 111, 111, 111, 111, 111, 112, 112, 112, 112, 112, 113, 113, 113, 113, 113, 114, 114, 114, 114, 114, 115, 115, 115, 115, 115, 116, 116, 116, 116, 116, 117, 117, 117, 117, 117, 118, 118, 118, 118, 118, 119, 119, 119, 119, 119, 120, 120, 120, 120, 120, 121, 121, 121, 121, 121, 122, 122, 122, 122, 122, 123, 123, 123, 123, 123, 124, 124, 124, 124, 124, 125, 125, 125, 125, 125, 126, 126, 126, 126, 126, 127, 127, 127, 127, 127, 128, 128, 128, 128, 128, 129, 129, 129, 129, 129, 130, 130, 130, 130, 130, 131, 131, 131, 131, 131, 132, 132, 132, 132, 132, 133, 133, 133, 133, 133, 134, 134, 134, 134, 134, 135, 135, 135, 135, 135, 136, 136, 136, 136, 136, 137, 137, 137, 137, 137, 138, 138, 138, 138, 138, 139, 139, 139, 139, 139, 140, 140, 140, 140, 140, 141, 141, 141, 141, 141, 142, 142, 142, 142, 142, 143, 143, 143, 143, 143, 144, 144, 144, 144, 144, 145, 145, 145, 145, 145, 146, 146, 146, 146, 146, 147, 147, 147, 147, 147, 148, 148, 148, 148, 148, 149, 149, 149, 149, 149, 150, 150, 150, 150, 150, 151, 151, 151, 151, 151, 152, 152, 152, 152, 152, 153, 153, 153, 153, 153, 154, 154, 154, 154, 154, 155, 155, 155, 155, 155, 156, 156, 156, 156, 156, 157, 157, 157, 157, 157, 158, 158, 158, 158, 158, 159, 159, 159, 159, 159, 160, 160, 160, 160, 160, 161, 161, 161, 161, 161, 162, 162, 162, 162, 162, 163, 163, 163, 163, 163, 164, 164, 164, 164, 164, 165, 165, 165, 165, 165, 166, 166, 166, 166, 166, 167, 167, 167, 167, 167, 168, 168, 168, 168, 168, 169, 169, 169, 169, 169, 170, 170, 170, 170, 170, 171, 171, 171, 171, 171, 172, 172, 172, 172, 172, 173, 173, 173, 173, 173, 174, 174, 174, 174, 174, 175, 175, 175, 175, 175, 176, 176, 176, 176, 176, 177, 177, 177, 177, 177, 178, 178, 178, 178, 178, 179, 179, 179, 179, 179, 180, 180, 180, 180, 180, 181, 181, 181, 181, 181, 182, 182, 182, 182, 182, 183, 183, 183, 183, 183, 184, 184, 184, 184, 184, 185, 185, 185, 185, 185, 186, 186, 186, 186, 186, 187, 187, 187, 187, 187, 188, 188, 188, 188, 188, 189, 189, 189, 189, 189, 190, 190, 190, 190, 190, 191, 191, 191, 191, 191, 192, 192, 192, 192, 192, 193, 193, 193, 193, 193, 194, 194, 194, 194, 194, 195, 195, 195, 195, 195, 196, 196, 196, 196, 196, 197, 197, 197, 197, 197, 198, 198, 198, 198, 198, 199, 199, 199, 199, 199, 200, 200, 200, 200, 200, 201, 201, 201, 201, 201, 202, 202, 202, 202, 202, 203, 203, 203, 203, 203, 204, 204, 204, 204, 204, 205, 205, 205, 205, 205, 206, 206, 206, 206, 206, 207, 207, 207, 207, 207, 208, 208, 208, 208, 208, 209, 209, 209, 209, 209, 210, 210, 210, 210, 210, 211, 211, 211, 211, 211, 212, 212, 212, 212, 212, 213, 213, 213, 213, 213, 214, 214, 214, 214, 214, 215, 215, 215, 215, 215, 216, 216, 216, 216, 216, 217, 217, 217, 217, 217, 218, 218, 218, 218, 218, 219, 219, 219, 219, 219, 220, 220, 220, 220, 220, 221, 221, 221, 221, 221, 222, 222, 222, 222, 222, 223, 223, 223, 223, 223, 224, 224, 224, 224, 224, 225, 225, 225, 225, 225, 226, 226, 226, 226, 226, 227, 227, 227, 227, 227, 228, 228, 228, 228, 228, 229, 229, 229, 229, 229, 230, 230, 230, 230, 230, 231, 231, 231, 231, 231, 232, 232, 232, 232, 232, 233, 233, 233, 233, 233, 234, 234, 234, 234, 234, 235, 235, 235, 235, 235, 236, 236, 236, 236, 236, 237, 237, 237, 237, 237, 238, 238, 238, 238, 238, 239, 239, 239, 239, 239, 240, 240, 240, 240, 240, 241, 241, 241, 241, 241, 242, 242, 242, 242, 242, 243, 243, 243, 243, 243, 244, 244, 244, 244, 244, 245, 245, 245, 245, 245, 246, 246, 246, 246, 246, 247, 247, 247, 247, 247, 248, 248, 248, 248, 248, 249, 249, 249, 249, 249, 250, 250, 250, 250, 250, 251, 251, 251, 251, 251, 252, 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892, 892, 892, 892, 893, 893, 893, 893, 893, 894, 894, 894, 894, 894, 895, 895, 895, 895, 895, 896, 896, 896, 896, 896, 897, 897, 897, 897, 897, 898, 898, 898, 898, 898, 899, 899, 899, 899, 899, 900, 900, 900, 900, 900, 901, 901, 901, 901, 901, 902, 902, 902, 902, 902, 903, 903, 903, 903, 903, 904, 904, 904, 904, 904, 905, 905, 905, 905, 905, 906, 906, 906, 906, 906, 907, 907, 907, 907, 907, 908, 908, 908, 908, 908, 909, 909, 909, 909, 909, 910, 910, 910, 910, 910, 911, 911, 911, 911, 911, 912, 912, 912, 912, 912, 913, 913, 913, 913, 913, 914, 914, 914, 914, 914, 915, 915, 915, 915, 915, 916, 916, 916, 916, 916, 917, 917, 917, 917, 917, 918, 918, 918, 918, 918, 919, 919, 919, 919, 919, 920, 920, 920, 920, 920, 921, 921, 921, 921, 921, 922, 922, 922, 922, 922, 923, 923, 923, 923, 923, 924, 924, 924, 924, 924, 925, 925, 925, 925, 925, 926, 926, 926, 926, 926, 927, 927, 927, 927, 927, 928, 928, 928, 928, 928, 929, 929, 929, 929, 929, 930, 930, 930, 930, 930, 931, 931, 931, 931, 931, 932, 932, 932, 932, 932, 933, 933, 933, 933, 933, 934, 934, 934, 934, 934, 935, 935, 935, 935, 935, 936, 936, 936, 936, 936, 937, 937, 937, 937, 937, 938, 938, 938, 938, 938, 939, 939, 939, 939, 939, 940, 940, 940, 940, 940, 941, 941, 941, 941, 941, 942, 942, 942, 942, 942, 943, 943, 943, 943, 943, 944, 944, 944, 944, 944, 945, 945, 945, 945, 945, 946, 946, 946, 946, 946, 947, 947, 947, 947, 947, 948, 948, 948, 948, 948, 949, 949, 949, 949, 949, 950, 950, 950, 950, 950, 951, 951, 951, 951, 951, 952, 952, 952, 952, 952, 953, 953, 953, 953, 953, 954, 954, 954, 954, 954, 955, 955, 955, 955, 955, 956, 956, 956, 956, 956, 957, 957, 957, 957, 957, 958, 958, 958, 958, 958, 959, 959, 959, 959, 959, 960, 960, 960, 960, 960, 961, 961, 961, 961, 961, 962, 962, 962, 962, 962, 963, 963, 963, 963, 963, 964, 964, 964, 964, 964, 965, 965, 965, 965, 965, 966, 966, 966, 966, 966, 967, 967, 967, 967, 967, 968, 968, 968, 968, 968, 969, 969, 969, 969, 969, 970, 970, 970, 970, 970, 971, 971, 971, 971, 971, 972, 972, 972, 972, 972, 973, 973, 973, 973, 973, 974, 974, 974, 974, 974, 975, 975, 975, 975, 975, 976, 976, 976, 976, 976, 977, 977, 977, 977, 977, 978, 978, 978, 978, 978, 979, 979, 979, 979, 979, 980, 980, 980, 980, 980, 981, 981, 981, 981, 981, 982, 982, 982, 982, 982, 983, 983, 983, 983, 983, 984, 984, 984, 984, 984, 985, 985, 985, 985, 985, 986, 986, 986, 986, 986, 987, 987, 987, 987, 987, 988, 988, 988, 988, 988, 989, 989, 989, 989, 989, 990, 990, 990, 990, 990, 991, 991, 991, 991, 991, 992, 992, 992, 992, 992, 993, 993, 993, 993, 993, 994, 994, 994, 994, 994, 995, 995, 995, 995, 995, 996, 996, 996, 996, 996, 997, 997, 997, 997, 997, 998, 998, 998, 998, 998, 999, 999, 999, 999, 999, 1000, 1000, 1000, 1000, 1000, 1001, 1001, 1001, 1001, 1001, 1002, 1002, 1002, 1002, 1002, 1003, 1003, 1003, 1003, 1003, 1004, 1004, 1004, 1004, 1004, 1005, 1005, 1005, 1005, 1005, 1006, 1006, 1006, 1006, 1006, 1007, 1007, 1007, 1007, 1007, 1008, 1008, 1008, 1008, 1008, 1009, 1009, 1009, 1009, 1009, 1010, 1010, 1010, 1010, 1010, 1011, 1011, 1011, 1011, 1011, 1012, 1012, 1012, 1012, 1012, 1013, 1013, 1013, 1013, 1013, 1014, 1014, 1014, 1014, 1014, 1015, 1015, 1015, 1015, 1015, 1016, 1016, 1016, 1016, 1016, 1017, 1017, 1017, 1017, 1017, 1018, 1018, 1018, 1018, 1018, 1019, 1019, 1019, 1019, 1019, 1020, 1020, 1020, 1020, 1020, 1021, 1021, 1021, 1021, 1021, 1022, 1022, 1022, 1022, 1022, 1023, 1023, 1023, 1023, 1023, 1024, 1024, 1024, 1024, 1024], [4.0, -1.0, -1.0, -1.0, -1.0, -1.0, 4.0, -1.0, -1.0, -1.0  …  -1.0, -1.0, -1.0, 4.0, -1.0, -1.0, -1.0, -1.0, -1.0, 4.0], 1024, 1024), [1.0 0.0 … 0.0 0.0; 0.0 1.0 … 0.0 0.0; … ; 0.0 0.0 … 1.0 0.0; 0.0 0.0 … 0.0 1.0], [0.0 0.03842943919353914 … 0.15224093497742697 0.03842943919353936; 0.03842943919353914 0.07685887838707828 … 0.1906703741709661 0.0768588783870785; … ; 0.15224093497742697 0.1906703741709661 … 0.30448186995485393 0.19067037417096633; 0.03842943919353914 0.07685887838707828 … 0.1906703741709661 0.0768588783870785])

One of the benefits of the Bayesian approach is that we can fit for the hyperparameters of our prior/regularizers unlike traditional RML appraoches. To construct this heirarchical prior we will first make a map that takes in our regularizer hyperparameters and returns the image prior given those hyperparameters.

fmap = let crcache=crcache
    x->GaussMarkovRandomField(x, 1.0, crcache)
end

#1 (generic function with 1 method)

Now we can finally form our image prior. For this we use a heirarchical prior where the correlation length is given by a inverse gamma prior to prevent overfitting. Gaussian Markov random fields are extremly flexible models. To prevent overfitting it is common to use priors that penalize complexity. Therefore, we want to use priors that enforce similarity to our mean image, and prefer smoothness.

cprior = HierarchicalPrior(fmap, InverseGamma(1.0, -log(0.01*rat)))

prior = NamedDist(c = cprior, σimg = truncated(Normal(0.0, 1.0); lower=0.01), fg=Uniform(0.0, 1.0))

lklhd = RadioLikelihood(sky, dlcamp, dcphase;
                        skymeta = metadata)
post = Posterior(lklhd, prior)

Posterior{RadioLikelihood{Comrade.ModelMetadata{typeof(Main.sky), NamedTuple{(:meanpr, :K, :grid, :cache), Tuple{Matrix{Float64}, VLBIImagePriors.CenterImage{Matrix{Float64}, Tuple{Int64, Int64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}, VLBISkyModels.NUFTCache{VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}, NFFT.NFFTPlan{Float64, 2, 1}, Vector{ComplexF64}, BSplinePulse{3}, KeyedArray{Float64, 2, NamedDimsArray{(:X, :Y), Float64, 2, Matrix{Float64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}}}}}}, Nothing, Tuple{Comrade.EHTObservation{Float64, Comrade.EHTLogClosureAmplitudeDatum{Float64}, StructArrays.StructVector{Comrade.EHTLogClosureAmplitudeDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :U4, :V4, :T, :F, :quadrangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{NTuple{4, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}, Comrade.EHTObservation{Float64, Comrade.EHTClosurePhaseDatum{Float64}, StructArrays.StructVector{Comrade.EHTClosurePhaseDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :T, :F, :triangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}}, Tuple{Comrade.ConditionedLikelihood{Comrade.var"#34#35"{Float64, Base.Fix2{typeof(logclosure_amplitudes), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}, Comrade.ConditionedLikelihood{Comrade.var"#36#37"{Float64, Base.Fix2{typeof(closure_phases), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, NamedTuple{(:U, :V, :T, :F), NTuple{4, Vector{Float64}}}}, VLBIImagePriors.NamedDist{(:c, :σimg, :fg), Tuple{VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}, Distributions.Uniform{Float64}}}}(RadioLikelihood
	Number of data products: 2
, VLBIImagePriors.NamedDist{(:c, :σimg, :fg), Tuple{VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}, Distributions.Uniform{Float64}}}(
dists: (VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}(
priormap: #1
hyperprior: Distributions.InverseGamma{Float64}(
invd: Distributions.Gamma{Float64}(α=1.0, θ=0.3410052124381751)
θ: 2.932506494109095
)

)
, Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.01), Distributions.Uniform{Float64}(a=0.0, b=1.0))
)
)

Reconstructing the Image

To sample from this posterior, it is convenient to first move from our constrained parameter space to an unconstrained one (i.e., the support of the transformed posterior is (-∞, ∞)). This is done using the asflat function.

tpost = asflat(post)

Comrade.TransformedPosterior{Posterior{RadioLikelihood{Comrade.ModelMetadata{typeof(Main.sky), NamedTuple{(:meanpr, :K, :grid, :cache), Tuple{Matrix{Float64}, VLBIImagePriors.CenterImage{Matrix{Float64}, Tuple{Int64, Int64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}, VLBISkyModels.NUFTCache{VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}, NFFT.NFFTPlan{Float64, 2, 1}, Vector{ComplexF64}, BSplinePulse{3}, KeyedArray{Float64, 2, NamedDimsArray{(:X, :Y), Float64, 2, Matrix{Float64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}}}}}}, Nothing, Tuple{Comrade.EHTObservation{Float64, Comrade.EHTLogClosureAmplitudeDatum{Float64}, StructArrays.StructVector{Comrade.EHTLogClosureAmplitudeDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :U4, :V4, :T, :F, :quadrangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{NTuple{4, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}, Comrade.EHTObservation{Float64, Comrade.EHTClosurePhaseDatum{Float64}, StructArrays.StructVector{Comrade.EHTClosurePhaseDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :T, :F, :triangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}}, Tuple{Comrade.ConditionedLikelihood{Comrade.var"#34#35"{Float64, Base.Fix2{typeof(logclosure_amplitudes), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}, Comrade.ConditionedLikelihood{Comrade.var"#36#37"{Float64, Base.Fix2{typeof(closure_phases), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, NamedTuple{(:U, :V, :T, :F), NTuple{4, Vector{Float64}}}}, VLBIImagePriors.NamedDist{(:c, :σimg, :fg), Tuple{VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}, Distributions.Uniform{Float64}}}}, TransformVariables.TransformTuple{NamedTuple{(:c, :σimg, :fg), Tuple{TransformVariables.TransformTuple{NamedTuple{(:params, :hyperparams), Tuple{TransformVariables.ArrayTransformation{TransformVariables.Identity, 2}, TransformVariables.ShiftedExp{true, Float64}}}}, TransformVariables.ShiftedExp{true, Float64}, TransformVariables.ScaledShiftedLogistic{Float64}}}}}(Posterior{RadioLikelihood{Comrade.ModelMetadata{typeof(Main.sky), NamedTuple{(:meanpr, :K, :grid, :cache), Tuple{Matrix{Float64}, VLBIImagePriors.CenterImage{Matrix{Float64}, Tuple{Int64, Int64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}, VLBISkyModels.NUFTCache{VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}, NFFT.NFFTPlan{Float64, 2, 1}, Vector{ComplexF64}, BSplinePulse{3}, KeyedArray{Float64, 2, NamedDimsArray{(:X, :Y), Float64, 2, Matrix{Float64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}}}}}}, Nothing, Tuple{Comrade.EHTObservation{Float64, Comrade.EHTLogClosureAmplitudeDatum{Float64}, StructArrays.StructVector{Comrade.EHTLogClosureAmplitudeDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :U4, :V4, :T, :F, :quadrangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{NTuple{4, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}, Comrade.EHTObservation{Float64, Comrade.EHTClosurePhaseDatum{Float64}, StructArrays.StructVector{Comrade.EHTClosurePhaseDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :T, :F, :triangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}}, Tuple{Comrade.ConditionedLikelihood{Comrade.var"#34#35"{Float64, Base.Fix2{typeof(logclosure_amplitudes), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}, Comrade.ConditionedLikelihood{Comrade.var"#36#37"{Float64, Base.Fix2{typeof(closure_phases), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, NamedTuple{(:U, :V, :T, :F), NTuple{4, Vector{Float64}}}}, VLBIImagePriors.NamedDist{(:c, :σimg, :fg), Tuple{VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}, Distributions.Uniform{Float64}}}}(RadioLikelihood
	Number of data products: 2
, VLBIImagePriors.NamedDist{(:c, :σimg, :fg), Tuple{VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}, Distributions.Uniform{Float64}}}(
dists: (VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}(
priormap: #1
hyperprior: Distributions.InverseGamma{Float64}(
invd: Distributions.Gamma{Float64}(α=1.0, θ=0.3410052124381751)
θ: 2.932506494109095
)

)
, Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.01), Distributions.Uniform{Float64}(a=0.0, b=1.0))
)
), TransformVariables.TransformTuple{NamedTuple{(:c, :σimg, :fg), Tuple{TransformVariables.TransformTuple{NamedTuple{(:params, :hyperparams), Tuple{TransformVariables.ArrayTransformation{TransformVariables.Identity, 2}, TransformVariables.ShiftedExp{true, Float64}}}}, TransformVariables.ShiftedExp{true, Float64}, TransformVariables.ScaledShiftedLogistic{Float64}}}}((c = TransformVariables.TransformTuple{NamedTuple{(:params, :hyperparams), Tuple{TransformVariables.ArrayTransformation{TransformVariables.Identity, 2}, TransformVariables.ShiftedExp{true, Float64}}}}((params = TransformVariables.ArrayTransformation{TransformVariables.Identity, 2}(asℝ, (32, 32)), hyperparams = as(Real, 0.0, ∞)), 1025), σimg = as(Real, 0.01, ∞), fg = as(Real, 0.0, 1.0)), 1027))

We can now also find the dimension of our posterior or the number of parameters we will sample.

ndim = dimension(tpost)

1027

Now we optimize using LBFGS

using ComradeOptimization
using OptimizationOptimJL
using Zygote
f = OptimizationFunction(tpost, Optimization.AutoZygote())
prob = Optimization.OptimizationProblem(f, prior_sample(rng, tpost), nothing)
sol = solve(prob, LBFGS(); maxiters=5_00);

Before we analyze our solution we first need to transform back to parameter space.

xopt = transform(tpost, sol)

(c = (params = [-0.8494620261149127 -0.6903432132227583 … -1.1460429104665082 -0.9962305690916786; -0.804566006281114 -0.634332548000261 … -1.283397291236575 -1.025321655049105; … ; -0.7968146139720274 -0.6670668747733154 … -0.9086999181453329 -0.8422587408717032; -0.8009085653610114 -0.6496714575767522 … -1.025180006630898 -0.9163586462477289], hyperparams = 19.018518327712204), σimg = 3.381626657616295, fg = 0.6883212825680473)

First we will evaluate our fit by plotting the residuals

using Plots
residual(skymodel(post, xopt), dlcamp, ylabel="Log Closure Amplitude Res.")

and now closure phases

residual(skymodel(post, xopt), dcphase, ylabel="|Closure Phase Res.|")

Now let's plot the MAP estimate.

import WGLMakie as CM
img = intensitymap(skymodel(post, xopt), μas2rad(150.0), μas2rad(150.0), 100, 100)
CM.image(img, axis=(xreversed=true, aspect=1, title="MAP Image"), colormap=:afmhot, figure=(;resolution=(400, 400),))

To sample from the posterior we will use HMC and more specifically the NUTS algorithm. For information about NUTS see Michael Betancourt's notes.

using ComradeAHMC
using Zygote
metric = DiagEuclideanMetric(ndim)
chain, stats = sample(post, AHMC(;metric, autodiff=Val(:Zygote)), 700; nadapts=500, init_params=xopt)

(NamedTuple{(:c, :σimg, :fg), Tuple{NamedTuple{(:params, :hyperparams), Tuple{Matrix{Float64}, Float64}}, Float64, Float64}}[(c = (params = [-0.8707367445791798 -0.7403260915175931 … -1.15651014364967 -0.9924354305530367; -0.7394866566190408 -0.6470282052996812 … -1.1640254728206514 -0.9761736763930722; … ; -0.8285140379046347 -0.6766135381588099 … -0.9661138230578891 -0.7984156417560282; -0.9106435153057657 -0.6030029865613113 … -1.0856987520875327 -0.9219187491549726], hyperparams = 20.98101502790318), σimg = 3.487996960826655, fg = 0.6941152691217748), (c = (params = [-0.8757158920000999 -1.0275632874112781 … -1.060821385133838 -1.4211365976553139; -0.3569521292016712 -0.4069531483603363 … -0.5935879828326397 -1.4760262273640081; … ; -1.0353841733990898 -0.5933677151468503 … -0.8995329458566653 -0.2083744111065022; -0.4557074645680682 -1.011939188731972 … -0.9521648544441836 -0.5554126927070474], hyperparams = 30.260720547532326), σimg = 2.7717989757054338, fg = 0.5782642382184245), (c = (params = [-0.22191679686189533 -0.1144718711123005 … -1.2512034104695982 -0.33383281158455613; -1.0290945979948416 -0.6116330497345576 … -2.5458874087059113 -0.9211912097900632; … ; 0.44221625743558934 0.3547054450257272 … -1.3979780541046571 -1.528570435421517; -0.40757891630441023 0.5589924461443299 … -1.4084766129084896 -1.0635244183197237], hyperparams = 26.464785087115395), σimg = 2.703853803859789, fg = 0.6323850090940326), (c = (params = [-0.15061970876177935 0.4948569465789101 … -2.4003869409728056 -2.0683080063361614; 0.41686129412900313 0.04815302525201166 … -1.1210405501095604 -1.3682923034417434; … ; -0.18743718473004534 1.0929036809912092 … -0.7952297402669114 0.07074706550105597; 1.2638983092324412 1.073468115698557 … -1.761297755295888 -0.7001833776859717], hyperparams = 33.22546133136862), σimg = 1.7714274332991184, fg = 0.4034293080475341), (c = (params = [-0.06381955186119659 0.4258900114807722 … -1.6387400361912368 -1.0392571315508121; -1.5963838898093083 -0.46710550696313863 … -1.4555343168480845 -1.5251827827693865; … ; 0.31334407614431237 0.13661193127735125 … -1.0978754756353042 -0.6418869908335816; 0.2365110884286247 1.167560207191675 … -0.9141272072128936 -0.5942169867016169], hyperparams = 15.50038715837744), σimg = 1.7093998535296993, fg = 0.5359164521212822), (c = (params = [-0.33262248892041607 0.3406516492533546 … -1.7196882924302088 -1.0461054946146149; -1.8001415762059194 -0.24320428296821747 … -1.2889608676742517 -1.7717285304538362; … ; 0.2665139780865846 0.2452019685450998 … -1.019916002583619 -0.525039906350776; 0.7642070354806949 1.3924847456356937 … -0.7476679106764408 -0.6388172872829413], hyperparams = 15.698030778225137), σimg = 1.7853194092440805, fg = 0.49452341835647556), (c = (params = [-0.8193052930807934 -1.2814972718725959 … -1.6389615685757053 -1.343911534269725; -1.7386115804562268 -1.9722996253441614 … -2.1998711802558044 -0.7337454011958177; … ; -0.3950831217471915 -0.25307948640691563 … -1.3898767353935249 -1.111351939289522; -1.3006294242774248 -1.1212759639847825 … -1.5474153812331193 -1.0029933912980855], hyperparams = 56.73615391610921), σimg = 1.878213250865731, fg = 0.43223551925173015), (c = (params = [-0.6486359046236648 -1.0913252781134004 … -2.170258515302875 -1.2203295845323991; -1.4314153297253924 -2.1591029093474607 … -2.4522588192004466 -0.8465687203031378; … ; -0.640430257293588 -0.8219921453884759 … -0.1812857524868295 -0.48934360175434205; -1.6999133116947247 -0.8008948785044255 … -1.6764032678365888 -1.311847448334331], hyperparams = 76.32180463245776), σimg = 1.8729577900690266, fg = 0.4087622502680217), (c = (params = [-0.6699730053198102 -1.0977264430901603 … -2.16723125514511 -1.2115581096019457; -1.4179634682555875 -2.169204975115125 … -2.445992137992968 -0.8517348719616724; … ; -0.6424803915969278 -0.8280289807994841 … -0.1813098854106631 -0.4845220201774133; -1.6792241038112172 -0.7941377763511521 … -1.6957101981466007 -1.3081301335942426], hyperparams = 74.93304136337287), σimg = 1.8550999700135293, fg = 0.41025575676524106), (c = (params = [-1.2546875076985353 -0.9341950876663733 … -0.2405839437502702 -0.6855127440449496; -0.5717570950770672 -0.40751980797937887 … -0.6678371544421967 -1.1261027952269491; … ; -0.8437297776267978 -0.7918032981826628 … -1.2217587521406001 -0.926133795421103; 0.011894487685682156 -1.12103856505103 … -0.4190321065314647 -0.4191105914186921], hyperparams = 11.716033410577653), σimg = 1.5766697215549974, fg = 0.47073185145389496)  …  (c = (params = [-0.00709572794022341 0.6345837755662059 … -0.8967838023993521 0.3090528799583476; 0.045089931501225866 0.8301447374953761 … -0.3584873434715615 -0.8317752248333539; … ; -0.16478628353576058 0.05798559336162153 … -0.6302006561122646 -0.4013318257412344; -0.31905310867200415 0.3964013617646047 … -0.9562394675285059 -0.10904951412935802], hyperparams = 8.052571085442912), σimg = 1.7159959899036947, fg = 0.2735140799943568), (c = (params = [0.6482823409352195 -0.04971958195029432 … 1.035749514925943 -0.03349558358092473; -0.05201804428566951 -1.0172320903957925 … -0.2826491302965055 0.8729222143943046; … ; 0.20874806324705536 0.5430656820490984 … 0.03846492629643203 0.20850599932447647; 0.7084210580916411 0.40383892825522555 … 0.5418334396142914 -0.07317100809575491], hyperparams = 12.3612541027683), σimg = 1.7778861650661804, fg = 0.25117706922101096), (c = (params = [0.3139255000350559 0.5033433998974821 … -0.0665989522135964 0.8695260415144255; -0.7359362492964278 0.10360841316498823 … -0.15506342281208185 -0.8008922150600354; … ; 0.4627906204160821 -0.40615925448795004 … 0.249233597179027 -0.1524408887213427; 0.030914582692538 0.013236543929509799 … -0.2273277476122592 -0.11475541847268446], hyperparams = 9.634904749653492), σimg = 1.6561556721796897, fg = 0.24143899113728454), (c = (params = [-1.3800195397767105 -0.6076250126341347 … 0.3078921038100295 -0.509467959030893; -1.2545150422239055 -0.5849660372223733 … 0.061775899088050326 -0.28223812328085146; … ; 0.21662923047879948 1.0135602779002464 … -0.6641861894338912 0.04841349138055748; -0.4186644082284004 -0.013301498976139458 … -0.4145127121017205 0.2425722904451215], hyperparams = 12.704927737468218), σimg = 1.5014238636133486, fg = 0.22679572188242003), (c = (params = [-0.575268594893293 -1.3571310925548479 … 0.15264298448334807 -0.19075309764889992; -0.422655824625412 -0.9037509128506621 … -0.4348822938254651 -0.35037930076718704; … ; 0.7221909839339692 1.3007169471488986 … -0.13162262998470733 0.31562240907603073; -0.14878217335349284 0.1321861835706768 … -0.35623990064243294 -0.24356772590803713], hyperparams = 5.8044851361613725), σimg = 1.7914324189554338, fg = 0.33371154188917673), (c = (params = [0.16762490504084135 -0.08264504120310731 … -0.20902219262785612 -0.1995468231090248; 0.08679569416763344 0.4845699094829764 … 0.3166558554180183 0.13605690940460438; … ; -0.5157747139674839 -0.4052895073069947 … -0.35782260862106874 -0.29788331102159876; -0.1258062210944784 -0.5296248117666699 … 0.2714471876754019 -0.16710097826814202], hyperparams = 10.863042114945953), σimg = 1.752332858021706, fg = 0.28890618926462464), (c = (params = [0.33537006831988 0.899757992210351 … -0.2828477579551726 0.1617577139742334; -0.06206673010476951 -0.007059241922238769 … -0.5626925313633698 -0.3593709558516884; … ; 0.8031463351951146 1.7245904110365373 … -1.0394797781601288 0.3763376002293546; 1.2624693418163258 1.5654773308890673 … -0.6868851788085477 -0.1672242154820182], hyperparams = 4.810104565111927), σimg = 1.7547821566226243, fg = 0.14381742859072588), (c = (params = [0.41978549495768946 1.0158057466222794 … 0.3761844815001938 0.2555130247024268; 0.43021917614700184 0.6693418362802429 … 0.6289766269045003 0.01897637098681562; … ; 0.5617648036211137 -0.3174089084744687 … 0.9614559229056834 0.3832105253218882; 0.31160187536550904 0.3390328767807594 … 0.8389509079702924 0.7668189288829399], hyperparams = 4.652846077162124), σimg = 1.7791348668915863, fg = 0.2384651432129868), (c = (params = [-0.26867584278199985 0.4678470002498427 … -0.2410854204856144 0.2560572828194496; 0.3502561970544669 0.9788229595665883 … -0.295493148611217 0.5349620013689237; … ; 1.312368561925051 1.623535426658103 … 0.4096091793574226 1.1302785610093264; 1.2793594014425298 1.0569134959530873 … 0.21932290994450238 0.7227123821952537], hyperparams = 3.646846022693653), σimg = 1.5199282549931707, fg = 0.156111083750984), (c = (params = [1.1101280444818178 0.7818102117568886 … 0.11774614738722121 0.46486094435216185; 0.4219388597506482 1.1206568926175202 … -0.19716683451503206 -0.24916369605862013; … ; 2.2198306796528047 1.1294044094364788 … 0.5974780227824257 0.4477927484563146; 1.6631518591124705 1.0343724732428152 … 0.2526362857705716 0.27379551146664227], hyperparams = 4.778428770594169), σimg = 1.6864921474018182, fg = 0.15085317867521772)], NamedTuple{(:n_steps, :is_accept, :acceptance_rate, :log_density, :hamiltonian_energy, :hamiltonian_energy_error, :max_hamiltonian_energy_error, :tree_depth, :numerical_error, :step_size, :nom_step_size, :is_adapt), Tuple{Int64, Bool, Float64, Float64, Float64, Float64, Float64, Int64, Bool, Float64, Float64, Bool}}[(n_steps = 1023, is_accept = 1, acceptance_rate = 0.9996986078628267, log_density = -477.07386097271035, hamiltonian_energy = 991.9453652621554, hamiltonian_energy_error = 0.0005478605208963927, max_hamiltonian_energy_error = 0.0006367788505485805, tree_depth = 10, numerical_error = 0, step_size = 0.0001, nom_step_size = 0.0001, is_adapt = 1), (n_steps = 1023, is_accept = 1, acceptance_rate = 0.9715600286314382, log_density = -721.137034938023, hamiltonian_energy = 986.8748863957882, hamiltonian_energy_error = 0.022703705553340114, max_hamiltonian_energy_error = -0.28438953267595934, tree_depth = 10, numerical_error = 0, step_size = 0.001574594011922962, nom_step_size = 0.001574594011922962, is_adapt = 1), (n_steps = 1023, is_accept = 1, acceptance_rate = 0.560683417342327, log_density = -741.9176846697401, hamiltonian_energy = 1257.0848632896698, hamiltonian_energy_error = -0.23548993926942785, max_hamiltonian_energy_error = 2.3024574984531228, tree_depth = 10, numerical_error = 0, step_size = 0.003036687425155042, nom_step_size = 0.003036687425155042, is_adapt = 1), (n_steps = 1023, is_accept = 1, acceptance_rate = 0.9547945805053729, log_density = -747.2560679635827, hamiltonian_energy = 1259.0101500940937, hamiltonian_energy_error = 0.05256709683635563, max_hamiltonian_energy_error = 0.1377591388916244, tree_depth = 10, numerical_error = 0, step_size = 0.0021197309544710124, nom_step_size = 0.0021197309544710124, is_adapt = 1), (n_steps = 511, is_accept = 1, acceptance_rate = 0.9673726218209104, log_density = -684.2392495336084, hamiltonian_energy = 1258.9351537555326, hamiltonian_energy_error = -0.006402006915777747, max_hamiltonian_energy_error = 0.12159172215979197, tree_depth = 9, numerical_error = 0, step_size = 0.004017565403384924, nom_step_size = 0.004017565403384924, is_adapt = 1), (n_steps = 36, is_accept = 1, acceptance_rate = 0.6143207247422399, log_density = -702.3850165343499, hamiltonian_energy = 1202.84555561722, hamiltonian_energy_error = 0.2548994607234363, max_hamiltonian_energy_error = 1232.3566774924016, tree_depth = 5, numerical_error = 1, step_size = 0.008160033006015446, nom_step_size = 0.008160033006015446, is_adapt = 1), (n_steps = 511, is_accept = 1, acceptance_rate = 0.6947579268043937, log_density = -650.7659740969779, hamiltonian_energy = 1199.1275501486903, hamiltonian_energy_error = 0.5490804890478103, max_hamiltonian_energy_error = 1.40951602504947, tree_depth = 9, numerical_error = 0, step_size = 0.005699932083801054, nom_step_size = 0.005699932083801054, is_adapt = 1), (n_steps = 511, is_accept = 1, acceptance_rate = 0.9652965251946474, log_density = -642.6048222314911, hamiltonian_energy = 1175.464872601537, hamiltonian_energy_error = -0.0032519488031539368, max_hamiltonian_energy_error = -0.6237644144291608, tree_depth = 9, numerical_error = 0, step_size = 0.0049400409622143565, nom_step_size = 0.0049400409622143565, is_adapt = 1), (n_steps = 5, is_accept = 1, acceptance_rate = 0.20000202580204304, log_density = -641.8070018799731, hamiltonian_energy = 1141.0143062178618, hamiltonian_energy_error = -1.3666291727004136, max_hamiltonian_energy_error = 2833.05242831509, tree_depth = 2, numerical_error = 1, step_size = 0.009869070749037484, nom_step_size = 0.009869070749037484, is_adapt = 1), (n_steps = 1023, is_accept = 1, acceptance_rate = 0.9787518927019421, log_density = -658.3552630384564, hamiltonian_energy = 1214.647916378201, hamiltonian_energy_error = 0.024763838674061844, max_hamiltonian_energy_error = 0.05528851087660769, tree_depth = 10, numerical_error = 0, step_size = 0.001757059105316145, nom_step_size = 0.001757059105316145, is_adapt = 1)  …  (n_steps = 255, is_accept = 1, acceptance_rate = 0.8085700177289067, log_density = -638.3654884538112, hamiltonian_energy = 1123.183552898027, hamiltonian_energy_error = 0.17895221109915838, max_hamiltonian_energy_error = 0.4530692346675096, tree_depth = 8, numerical_error = 0, step_size = 0.011104367630799094, nom_step_size = 0.011104367630799094, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.9888803705809813, log_density = -668.9305874843261, hamiltonian_energy = 1180.2449988526885, hamiltonian_energy_error = -0.20375286621333544, max_hamiltonian_energy_error = -0.2876292756025123, tree_depth = 8, numerical_error = 0, step_size = 0.011104367630799094, nom_step_size = 0.011104367630799094, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.938907099981244, log_density = -630.0776975268662, hamiltonian_energy = 1172.8384301340398, hamiltonian_energy_error = 0.0649211455713612, max_hamiltonian_energy_error = 0.20669624090692196, tree_depth = 8, numerical_error = 0, step_size = 0.011104367630799094, nom_step_size = 0.011104367630799094, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.6984435692380154, log_density = -652.273087114519, hamiltonian_energy = 1165.9750349176854, hamiltonian_energy_error = 0.4150181877018895, max_hamiltonian_energy_error = 0.8399031351314079, tree_depth = 8, numerical_error = 0, step_size = 0.011104367630799094, nom_step_size = 0.011104367630799094, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.981890250683445, log_density = -638.5821568861345, hamiltonian_energy = 1177.3896646749763, hamiltonian_energy_error = -0.3153273308748794, max_hamiltonian_energy_error = -0.7083789681141752, tree_depth = 8, numerical_error = 0, step_size = 0.011104367630799094, nom_step_size = 0.011104367630799094, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.9721378348155116, log_density = -630.3633495323957, hamiltonian_energy = 1179.1777378676397, hamiltonian_energy_error = -0.06856841178705508, max_hamiltonian_energy_error = -0.5654754100664832, tree_depth = 8, numerical_error = 0, step_size = 0.011104367630799094, nom_step_size = 0.011104367630799094, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.8969840071309876, log_density = -616.2364094204758, hamiltonian_energy = 1124.6852324356996, hamiltonian_energy_error = 0.21179412629044236, max_hamiltonian_energy_error = 0.4794545660431595, tree_depth = 8, numerical_error = 0, step_size = 0.011104367630799094, nom_step_size = 0.011104367630799094, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.7883469249118109, log_density = -618.5905030765854, hamiltonian_energy = 1119.4853607658881, hamiltonian_energy_error = 0.10995517456194648, max_hamiltonian_energy_error = 1.1975911042691223, tree_depth = 8, numerical_error = 0, step_size = 0.011104367630799094, nom_step_size = 0.011104367630799094, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.9746675969959911, log_density = -587.9956227234836, hamiltonian_energy = 1084.7521144109564, hamiltonian_energy_error = -0.3314492211259221, max_hamiltonian_energy_error = -1.7110871227816915, tree_depth = 8, numerical_error = 0, step_size = 0.011104367630799094, nom_step_size = 0.011104367630799094, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.9928633114041997, log_density = -617.1084537000218, hamiltonian_energy = 1093.838419727731, hamiltonian_energy_error = -0.1520871070042631, max_hamiltonian_energy_error = -0.5309602007041576, tree_depth = 8, numerical_error = 0, step_size = 0.011104367630799094, nom_step_size = 0.011104367630799094, is_adapt = 0)])

Now that we have our posterior, we can assess which parts of the image are strongly inferred by the data. This is rather unique to Comrade where more traditional imaging algorithms like CLEAN and RML are inherently unable to assess uncertainty in their reconstructions.

To explore our posterior let's first create images from a bunch of draws from the posterior

msamples = skymodel.(Ref(post), chain[501:2:end]);

The mean image is then given by

using StatsBase
imgs = intensitymap.(msamples, μas2rad(150.0), μas2rad(150.0), 128, 128)
mimg = mean(imgs)
simg = std(imgs)
fig = CM.Figure(;resolution=(400, 400));
CM.image(fig[1,1], mimg,
                   axis=(xreversed=true, aspect=1, title="Mean Image"),
                   colormap=:afmhot)
CM.image(fig[1,2], simg./(max.(mimg, 1e-5)),
                   axis=(xreversed=true, aspect=1, title="1/SNR",), colorrange=(0.0, 2.0),
                   colormap=:afmhot)
CM.image(fig[2,1], imgs[1],
                   axis=(xreversed=true, aspect=1,title="Draw 1"),
                   colormap=:afmhot)
CM.image(fig[2,2], imgs[end],
                   axis=(xreversed=true, aspect=1,title="Draw 2"),
                   colormap=:afmhot)
fig

Now let's see whether our residuals look better.

p = plot();
for s in sample(chain[501:end], 10)
    residual!(p, vlbimodel(post, s), dlcamp)
end
ylabel!("Log-Closure Amplitude Res.");
p

p = plot();
for s in sample(chain[501:end], 10)
    residual!(p, vlbimodel(post, s), dcphase)
end
ylabel!("|Closure Phase Res.|");
p

And viola, you have a quick and preliminary image of M87 fitting only closure products. For a publication-level version we would recommend

1. Running the chain longer and multiple times to properly assess things like ESS and R̂ (see Geometric Modeling of EHT Data)
2. Fitting gains. Typically gain amplitudes are good to 10-20% for the EHT not the infinite uncertainty closures implicitly assume
3. Making sure the posterior is unimodal (hint for this example it isn't!). The EHT image posteriors can be pretty complicated, so typically you want to use a sampler that can deal with multi-modal posteriors. Check out the package Pigeons.jl for an in-development package that should easily enable this type of sampling.

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